I believe the most accurate answer would be that the United States has what is currently the longest running constitution in the world, marking 219 years this coming Sept. 17.
Actually, many consider the U.S. Constitution the world's first constitution, setting up very defined rules of how government was to be established and rules defining its method of operation.
But, as an earlier respondent mentioned, many past groups, such as native American tribes and even the Romans, Greeks, Egyptians, Sumarians and others had codified agreements on how ruling was to be done that could liberally be called constitutions... even if those rules did revolve around forms of monarchs that, in the end, had absolute power in all matters.
But since none of those groups (expept perhaps the Greeks and Egyptians) could be currently said to exist in a formally defined and internationally recognized nation, any long term streaks under those "constitutions" would not be current.